文摘
We investigate several problems in the theory of convergence spaces: generalization of Kolmogorov separation from topological spaces to convergence spaces,representation of reflexive digraphs as convergence spaces,construction of differential calculi on convergence spaces,mereology on convergence spaces,and construction of a universal homogeneous pretopological space. First,we generalize Kolmogorov separation from topological spaces to convergence spaces; we then study properties of Kolmogorov spaces. Second,we develop a theory of reflexive digraphs as convergence spaces,which we then specialize to Cayley graphs. Third,we conservatively extend the concept of differential from the spaces of classical analysis to arbitrary convergence spaces; we then use this extension to obtain differential calculi for finite convergence spaces,finite Kolmogorov spaces,finite groups,Boolean hypercubes,labeled graphs,the Cantor tree,and real and binary sequences. Fourth,we show that a standard axiomatization of mereology is equivalent to the condition that a topological space is discrete,and consequently,any model of general extensional mereology is indistinguishable from a model of set theory; we then generalize these results to the cartesian closed category of convergence spaces. Finally,we show that every convergence space can be embedded into a homogeneous convergence space; we then use this result to construct a universal homogeneous pretopological space.